Finite intervals of multisets

This file provides the LocallyFiniteOrder instance for Multiset α and calculates the cardinality of its finite intervals.

Implementation notes

We implement the intervals via the intervals on DFinsupp, rather than via filtering Multiset.Powerset; this is because (Multiset.replicate n x).Powerset has 2^n entries not n+1 entries as it contains duplicates. We do not go via Finsupp as this would be noncomputable, and multisets are typically used computationally.

instance Multiset.instLocallyFiniteOrder {α : Type u_1} [DecidableEq α] : LocallyFiniteOrder (Multiset α)

Equations

• One or more equations did not get rendered due to their size.

theorem Multiset.Icc_eq {α : Type u_1} [DecidableEq α] (s t : Multiset α) : Finset.Icc s t = Finset.map Multiset.equivDFinsupp.symm.toEmbedding (Finset.Icc (Multiset.toDFinsupp s) (Multiset.toDFinsupp t))

theorem Multiset.uIcc_eq {α : Type u_1} [DecidableEq α] (s t : Multiset α) : Finset.uIcc s t = Finset.map Multiset.equivDFinsupp.symm.toEmbedding (Finset.uIcc (Multiset.toDFinsupp s) (Multiset.toDFinsupp t))

theorem Multiset.card_Icc {α : Type u_1} [DecidableEq α] (s t : Multiset α) : (Finset.Icc s t).card = ∏ i ∈ s.toFinset ∪ t.toFinset, (Multiset.count i t + 1 - Multiset.count i s)

theorem Multiset.card_Ico {α : Type u_1} [DecidableEq α] (s t : Multiset α) : (Finset.Ico s t).card = ∏ i ∈ s.toFinset ∪ t.toFinset, (Multiset.count i t + 1 - Multiset.count i s) - 1

theorem Multiset.card_Ioc {α : Type u_1} [DecidableEq α] (s t : Multiset α) : (Finset.Ioc s t).card = ∏ i ∈ s.toFinset ∪ t.toFinset, (Multiset.count i t + 1 - Multiset.count i s) - 1

theorem Multiset.card_Ioo {α : Type u_1} [DecidableEq α] (s t : Multiset α) : (Finset.Ioo s t).card = ∏ i ∈ s.toFinset ∪ t.toFinset, (Multiset.count i t + 1 - Multiset.count i s) - 2

theorem Multiset.card_uIcc {α : Type u_1} [DecidableEq α] (s t : Multiset α) : (Finset.uIcc s t).card = ∏ i ∈ s.toFinset ∪ t.toFinset, ((↑(Multiset.count i t) - ↑(Multiset.count i s)).natAbs + 1)

theorem Multiset.card_Iic {α : Type u_1} [DecidableEq α] (s : Multiset α) : (Finset.Iic s).card = ∏ i ∈ s.toFinset, (Multiset.count i s + 1)